Download 14.02 Quiz 1 Solution

14.02 Quiz 1 Solution
QUESTION 1: TRUE OR FALSE. Explain your answer fully 30 points (6 pts. each)
1.
Assume that government spending is a linear function of total output: i.e.
G = g0 - g1 Y, where g0 is positive. The Keynesian multiplier is larger when g1 is
positive than when g1 is equal to zero.
Solution. False: the equilibrium on the goods market implies that Y = c0 + c1 (Y –
T) + g0 – g1G or Y = 1/(1 – c1 + g1 )*(c0 + g0 – c1 T)The Keynesian multiplier is 1/(1 – c1
+ g1 ) which is smaller than 1/(1-c1 ). The positive impact of an increase in autonomous
spending is partially offset by the subsequent drop in government spending.
2. In an economy where individuals demand half of their money as currency and
half as checkable deposits, an increase in high-powered money by the Central
Bank has a larger effect on interest rates and output than in an economy where
individuals hold all of their money as cash.
Solution. True. The money multiplier in an economy where individuals demand half of
their money in currency and half as checkable deposits is 1/(.5+.5θ) which is higher than
1 as it would be in an economy where individuals would hold all their money as cash.
Therefore an increase in high-powered money by the Central Bank has a larger effect
(provided that banks do not keep all their money in reserves, θ<1, in which case the
money multiplier would always be 1).
3. The Bank of Japan (BoJ: central bank of Japan) has published the following
nominal GDP growth figures for 2000, 2001 and 2002:g2000 = 1.1%, g2001 = -2.5%
and g2002 = -0.7%. In the same publication the real GDP growth figures are: gr 2000
= 3.2%, gr 2001 = -1.2% and gr 2002 = 1.6%. These figures imply that the rate of
inflation was negative in 2000, positive in 2001 and again negative in 2002.
Solution. False: the rate of inflation will be positive if nominal output grows faster than
real output, and negative otherwise. In every year, the nominal GDP growth was lower
than the real GDP growth, implying that inflation was negative all three year (and not
positive in 2001 as stated). Japan experienced deflation three years in a row.
4. Fiscal policy cannot affect the level of GDP if money demand does not depend on
the interest rate
Solution. True: when money demand does not depend on the interest rate, the LM curve
is vertical, meaning that output is set on the financial market, no matter what the
conditions on the goods market are. Fiscal policy moves the IS curve, and this will have
no effect on output with a vertical LM curve: any increase in demand due to increase in
government spending is totally offset by an increase in the interest rate (which depresses
investment).
5. Consider the simple goods market model, where Z = C + I + G and both I and G
are fixed. Let the consumption function be C = c0 + c1 (Y-T). If c1 rises, the private
saving rate, defined as S/Y, decreases.
Solution. True: if c1 increases, the equilibrium output increases. On the other hand,
private savings must be equal to investment minus public savings, which is fixed.
Therefore, private savings are not affected (the savings paradox). If S is unchanged and Y
goes up, then S/Y decreases.
QUESTION 2: THE GOODS MARKET. 20 points (5 pts. each)
Consider the following equations describing the components of demand and equilibrium
in the goods market:
C= 120 + 0.5 (Y - T)
I = 40
G=20
T= 40
1. Solve for the equilibrium level of output in this economy.
Solution: The aggregate demand of this economy is given by
Z=C+I+G
Z = 120 + 0.5 (Y – T) + I + G
The equilibrium condition is that aggregate demand is equal to output: Z = Y
Y* = 120 + 0.5 (Y* – T) + I + G
Y* = 1/(1 – 0.5) (120 + I + G – 0.5T)
Y* = 2 (120 + 40 + 20 – 0.5*40)
Y* = 320
2. There is a permanent increase in government spending of 10 units (so, now
G=30). Solve for the new equilibrium Z and Y.
Solution: In this very simple goods market model, an increase in government spending of
10 units leads to an increase in equilibrium output of 10*1/(1-0.5) units (10 units times
the Keynesian multiplier).
Y* = 320 + 10*1/(1-0.5) = 320 + 20 = 340
Since we must have aggregate demand equal to output , Z = Y* = 340
Now, assume that it takes one period for the firms to adjust production in order to meet
demand. In this economy, the equilibrium condition is Yt =Zt-1 .
3. Assume that Y0 = Y-1 is equal to the equilibrium output you found in part 1. At
time t = 0, there is a permanent increase in government spending by 10 units (so,
now G=30). Solve for Z0 .
Solution: At time t = 0, government spending increases. Aggregate demand therefore
increases:
Z0 = C0 + I + G + 10
Z0 = 120 + 0.5(Y-1 – T) + I + G +10
Z0 = 120 + 0.5(320 – 40) + 40 + 20 + 10
Z0 = 330
The increase in government spending leads to a one for one increase in the aggregate
demand.
4. Solve for Y1 and Z1 and Y2 .
Solution: We know that Y1 = Z0 = 330.
Z1 = C1 + I + G
Z1 = 120 + 0.5(Y0 - T) + I + G +10
Z1 = 120 + 0.5(330 – 40) + 40 + 20 + 10
Z1 = 335
We know that Y2 = Z1 = 335.
The increase in aggregate demand at t = 1 leads to an increase in output. This implies
an increase in disposable income. Out of this additional disposable income, people
consume a fraction equal to their marginal propensity to consume (0.5). This leads to
a further increase in aggregate demand, and a subsequent increase in output.
Bonus question (10 points):
Solve for Yn and for Y`. Compare this result to your answer in question 2.
Solution: the first increment in output is 10 units. The second one is 0.5*10. The third
one will be 10*0.52 . The n-th one will be 10*0.5n . The total increase after n periods
will be:
Yn = 10*(1 + 0.52 + 0.53 + … + 0.5n )
Yn = 10*(1 – 0.5n+1 )/(1 – 0.5)
As the number of periods goes to infinity, 0.5n+1 goes to zero, and therefore the total
increment in output following a permanent increase of government spending of 10
units will be:
Y` = 1/(1-0.5)*10 = 20
This is the same increase as in question 2. The increase is equal to the increase in
government spending, 10 units, multiplied by the Keynesian multiplier 1/(1 – 0.5).
The Keynesian multiplier comes from the addition of all the successive rounds of
increase in demand (through increases in income and therefore of consumption).
QUESTION 3: IS-LM 50 points
Consider an economy described by the following short run model.
C = c1 (Y-T)
I = b0 + b1 Y- b2 i
Md /P = C-a*i
G = G0
T = T0
Ms = M0
All the parameters (a, b0 ,b1 , b2 , c1 ) in the model are positive, and b1 + c1 <1. We assume
for simplicity that P=1.
Important: Note that unlike in the short run model we have seen in class, money demand
depends on consumption, and not on total output.
Important : always give a brief explanation of what you are doing (for all the questions).
1. Find the equation for the aggregate demand (Z). Draw the aggregate demand
curve and the equilibrium in the goods market (45 degree line) on a graph with
output (Y) on the horizontal axis, and aggregate demand (Z) on the vertical axis.
Z= C + I + G = c1 (Y-T0 ) + b0 + b1 Y- b2 i + G0 = (G0 -c1 T0 +b0 -b2 i) + (c1 +b1 ) Y
Supply curve
Z= (G 0-c1T+b0-b2i) +(c 1+b1)Y
Z*
G0-c1T+b0-b2i
c1+b1
Equilibrium in the
goods market: Y=Z
45%
Y*
Y≡GDP
2. Find the equilibrium on the goods market, and derive the equation for the IS
curve.
In equilibrium: Y = Z ⇒ Y = (G 0 -c1 T0 +b0 -b2 i) + (c1 +b1 )Y
⇒i =
1
(G0 + b0 − c1T0 ) − (1 − (c1 + b1 ) ) Y ≡ a1 − a 2Y
b2
b2
⇔Y =
1
b2
(G0 + b0 − c1T0 ) −
i
1 − (c1 + b1 )
1 − ( c1 + b1 )
3. Find the equilibrium on the financial market. Derive the equation for the LM
curve.
Md = C-a*i = c1 (Y-T0 ) -a*i
Ms = M0
In equilibrium: Ms = Md
⇒ M0 = c1 (Y- T0 ) -a*i
1
(M 0 + c1T0 ) + c1 Y ≡ a3 + a4Y
a
a
1
a
⇔ Y = ( M 0 + c1T0 ) + i
c1
c1
⇒i =−
4. Show graphically the equilibrium in both the goods market and the financial
markets. Call the equilibrium output Y0 , the IS curve IS0 and the LM curve LM0 .
Do not do any algebra, but make sure you put all the available information on
your graph.
The IS curve gives the pairs of (Y,i) that support equilibrium in the goods market.
The LM curve gives the pairs of (Y,i) that support equilibrium in the money
market.
The intersection of both gives (Y,i) that support a simultaneous equilibrium in
both market.
i
Equilibrium in the goods
and money markets
a1
i*
a3
-a2
LM1 : i = a3 + a4 Y
A
IS1 : i = a1 – a2 Y
a4
Y*
a1/a2
Y≡GDP
5. There is a sudden drop in business confidence, and firms decide suddenly to
invest less for any level of output and interest rates. From this question onward,
the new behavior of firms obeys the following equation:
I = b0 - ? + b 1 Y- b2 i with ?>0
Find the new IS relation (call it IS1 ) and the new LM relation (call it LM1 ). Show
graphically how the two curves shift, and state what happens to equilibrium
output and interest rates. Call the new equilibrium output Y1 .
The IS relation is affected through the reduction in b0 , which affects the intersection
(↓a1), and causes the IS curve to shift left-down (? ):
⇒i =
1
(G0 + b0 − ∆ − c1T0 ) − (1 − ( c1 + b1 ) )Y ≡ a1' − a 2Y
b2
⇔Y =
1
b2
(G0 + b0 − ∆ − c1T0 ) −
i
1 − (c1 + b1 )
1 − ( c1 + b1 )
The LM curve is not affected.
Therefore, in the new equilibrium (Y↓,i↓), as shown in the following graph:
i
a1
New equilibrium in the
goods and money markets
i*
i**
LM1 : i = a3 + a4 Y
A
B
a3
IS2
Y** Y*
IS1 : i = a1 – a2 Y
a1/a2
Y≡GDP
6. Now suppose the government wants to counterbalance the effect of the drop in
investors’ confidence on GDP using government spending as a policy
instrument. Should the governme nt increase or decrease G? What new level of
government spending G1 should the government choose to completely offset the
effect of lower business confidence?
A change in G affects only the IS curve through its effect on the intersection (a1 ). The
government can counterbalance the effect of the drop in investors’ confidence on GDP by
increasing G by exactly ? (i.e., by the same drop in b0 ), which will shift IS back to its
original position:
⇒i =
1
(G0 + ∆ + b0 − ∆ − c1T0 ) − (1 − ( c1 + b1 ) )Y ≡ a1' − a 2Y
b2
⇒i =
1
(G0 + b0 − c1T0 ) − (1 − (c1 + b1 ) )Y ≡ a1 − a 2Y
b2
7. Suppose now the government cannot use fiscal instruments (G or T) because
Congress would not allow it. The Central Bank decides to use monetary policy to
restore the original equilibrium output Y0 .
Should the Central Bank increase or decrease Ms? Draw the new IS (call it IS2 )
and the new LM curve (call it LM 2 ) after the change in money supply.
M should be increased, so the LM curve shifts down-right (? ) and crosses IS at Y=Y*,
which is at point D in the graph below:
i
a1
New equilibrium in the
goods and money markets
LM1 : i = a3 + a4 Y
LM 2
i*
i**
A
B
a3
D
IS2
Y** Y*
IS1 curve: i = a1 – a2 Y
a1/a2
Y≡GDP
8. For each of the two sub-questions 6 and 7, state qualitatively how i and I move,
compared to sub-question 4, in response to the change in economic policy:?
(Qualitatively means whether a variable goes up, down, or stays unchanged
relative to sub-question 4)
A government expansion (section 6): The interest rate returns to its value in section 4, as
it is clear from the graph of section 6. By considering the investment equation, and noting
that the interest rate and GDP are the same in section 4 and 6, however the constant in
the investment equation is lower in section 6, then it must be that the investment is section
6 is lower than in section 4. Here government investment replaces private investment.
A monetary expansion: by inspection the graph in section 7, we see that the interest rate
is lower in section 6 than in section 4. So the investment will be higher than in section 6.
By how much? We know that in equilibrium Y = C + I + G. Y is the same as in question
4, C as well (same disposable income), and G as well. Therefore, I must be the same as in
question 4: the drop in interest rate has convinced the investors to go back to their
original level of investment.
9. Suppose the government wants both to return to the original level of output Y0
and to achieve as high a level of investment as possible. Of the two options
considered in sub-questions 6 and 7 (government spending or monetary policy)
which econo mic policy instrument should the government use? Briefly justify
your answer.
Since in both cases, the economy returns to the same level of Y, the government should
look for the policy with the lowest interest rate in the new equilibrium. Comparing the
above graphs, we find that monetary expansion brings about the lowest interest rate.
With a monetary expansion, investment is ? higher than with a fiscal expansion, and
output is the same.
10. Suppose the government decides to increase taxes. What is the effect on
equilibrium output and equilibrium interest rate? Draw the new IS curve (call it
IS3 ) and the new LM curve (call it LM 3 ). You must justify your answer.
A change in T affects both IS and LM:
IS: ↑T→ ↓a1 → shifts the IS curve up-right (?):
⇒i=
(
)
1
G 0 + b 0 − c1T0 ↑ − (1 − (c1 + b1 ))Y ≡↓ a1 − a 2Y
b2
LM: ↑T→ ↓a3 → shifts the LM curve up-left (? ).
⇒i=−
1
c
(
M 0 + c1T0 ↑ ) + 1 Y ≡↓ a3 + a4Y
a
a
In equilibrium: IS=LM
⇒
(
)
(
)
1
1
c
G0 + b0 − c1T0 ↑ − (1 − (c1 + b1 ))Y = − M 0 + c1T0 ↑ + 1 Y
b2
a
a
(
)
(
1
1
c

⇒  1 + 1 − (c1 + b1 ) Y =
G 0 + b 0 − c1T0 ↑ + M 0 + c1T0 ↑
b2
a
a

1
 1 1
1
c

⇒  1 + 1 − (c1 + b1 ) Y =  (G0 + b0 ) + M 0  +  −
a
a

 b2
  a b2
)

 c1T0 ↑

1

1 1 
1
 (G 0 + b0 ) + M 0 
 −  c1
b2
a


 a b2 
⇒Y =
+
T0 ↑≡ δ 0 + δ 1T0 ↑
 c1

 c1

 + 1 − (c1 + b1 ) 
 + 1 − (c1 + b1 ) 
a

a

Note that the sign of δ 1 depends on the relative values of a and b2 . if the sign of δ 1 is
positive, that is if b2 higher than a, then the intersection of the two curves will be moving
to the right as we increase taxes. Therefore, in this case, we need to increase taxes to
restore the original GDP, and the new equilibrium interest rate will be lower than the
original (see the graph below for this case). The opposite when the sign is negative.
The intuition is simple: if b2 is high compared to a, then changing taxes affects the IS
curve a lot (investment reacts to changes in interest rate), but the demand for money
reacts relatively little (little impact of changes in consumption on money demand) and the
LM curve does not move much.
i
a1
New equilibrium in the
goods and money markets
LM1 : Y = a3 + a4 i
C
i*
i**
A
LM 3
B
a3
i***
IS3
Y**
Y*
IS2
IS1 curve: Y = a1 – a2 i
a1/a2
Y≡GDP